130 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS NOTE B, Article 65. Let us resume the problem of Art. 7, but instead of the motion of the plane being periodic, let us suppose that the plane and fluid are initially at rest, and that the plane is then moved with a constant velocity V, and let the notation be the same as in Art. 7. The general equations (8) remain the same as before, but the equations of condition become in this case v = 0 when t = 0 from x = 0 to # = oo , v = Y when x = 0 from t = 0 to £ = oo . By Fourier's theorem and another theorem of the same kind, v may be expanded between the limits 0 and oo of x in the following form : : 2 TOO /*oo i) = - TrJo Jo cos ax cos CLX' 6 (#', t) dx'da 2 c°° c°° -f -I sin ax sin asc'ty (xf, t) dxda. ...... (184). 7T Jo Jo In fact, v could be expanded by means of either of these expressions separately, and of course can be expanded in an infinite number of ways by the sum of the two. If however v had been expanded by means of the first expression alone, its derivatives with respect to x could not have been obtained by differentiating under the integral signs, inasmuch as the derivatives of an odd order do not vanish when x = 0, but would have been given by certain formulae which I have investigated in a former paper*. A similar remark applies to the second expansion, in consequence of the circumstance that v itself and its derivatives of an even order do not vanish with x. But by combining the two expansions we may obtain the derivatives of v, up to any order i that we please to fix on, by merely differentiating under the integral signs. For we may evidently express the finite function v, and that in an infinite number of ways, as the sum of two finite functions <j> (x, t), ty(oG, t) which like v vanish when x = oo , and which are such that the odd derivatives of the first, and the even derivatives of the * On the critical values of the sums of periodic series. Garni. Phil. Trans., Vol. vin. p. 533. [Ante, Vol. i. p. 287.]